Chapter 7
Hypothesis Testing
7-1. H0: p = 0.8
H1: p ¹ 0.8
7-2. H0: m £ 200
H1: m > 200
7-3. H0: m £ 12
H1: m > 12
7-4. H0: m ³ $49,160
H1: m < $49,160
7-5. H0: m £ $1.78
H1: m > $1.78
7-6. The power of a test is the probability that a false null hypothesis will be detected by the test. It is important if the ramifications of accepting a false null hypothesis outweigh the costs of rejecting the null hypothesis.
7-7. Since a and b are inversely related and since power = (1 - b), then the level of significance a and the power of the test are directly related to each other: as a increases, power increases.
7-8. Increasing the level of significance.
7-9. a) Null: there are no weapons Alternative: there are weapons
b) Type I error: rejecting the null hypothesis and concluding there are weapons when there are no weapons.
Type II error: accepting the null hypothesis of no weapons when weapons are present.
c) Type II error would be more costly in terms of endangering lives. Type I would just be an inconvenience for the passenger.
d) Largest value of a possible.
e) If the sensitivity of the detectors is increased, it should result in more Type I errors and less Type II errors.
f) If a is to be increased, then the sensitivity of the detectors should also be increased to compensate for the higher a.
7-10. Plan for a large sample.
7-11. a) left tailed H1: m < 10
b) right tailed H1: p > 0.5
c) left tailed H1: m < 100
d) right tailed H1: m > 20
e) two-tailed H1: p ¹ 0.22
f) right tailed H1: m > 50
g) two-tailed H1: s2 ¹ 140
7-12. a) p= .5 - .4599 = 0.0401
b) p = 0.9599
c) p = .0802
7-13. a) to the left tail
b) to the right tail
c) either to the left or to the right tail
7-14. A test statistic is a sample statistic computed from the data. Knowing the distribution of the test statistic enables the construction of the rejection regions about the mean of the test statistic.
7-15. a) p value will decrease
b) p value increases
c) p value decreases
7-16. H0: = 31.5 H1: 31.5
n = 100 = 29.8 s = 6.6
z = = = -2.575
Reject H0 at = 0.05 (and almost at = .01). The p-value is approximately 0.01
7-17. H0: = 247 H1: 247
n = 60 = 250 s = 12 = 0.05, then also 0.01
z = = = 1.936
Do not reject H0 at = .05 nor at = .01. (p-value = .0528). Maybe take a larger sample, or increase your .
EvidenceSample size / 60 / n
Sample Mean / 250 / x-bar
s Known; Normal Population or Sample Size >= 30
Population Stdev. / 12 / s
Test Statistic / 1.9365 / z
At an a of
Null Hypothesis / p-value / 5%
H0: m = / 247 / 0.0528
7-18. H0: = 5 H1: 5
= .05 n = 120 = 2.3 s = 1.5
z = = -19.72
Reject H0 at = .05 and at smaller levels. Average miles traveled per day is probably smaller than 5. Changes in service may be necessary (the p-value is very small).
7-19. H0: = 77 H1: 77
n = 350 = 84 s = 28
z = = 4.68
Reject H0 at the .01 and even smaller levels of (the p-value is small). Customer satisfaction has probably improved.
7-20. H0: = 11.5 H1: 11.5
n = 50 = 10.8 s = 3.4 = .05
z = = -1.456
Do not reject H0 at any (p-value = 0.1454).
EvidenceSample size / 50 / n
Sample Mean / 10.8 / x-bar
s Known; Normal Population or Sample Size >= 30
Population Stdev. / 3.4 / s
Test Statistic / -1.4558 / z
At an a of
Null Hypothesis / p-value / 5%
H0: m = / 11.5 / 0.1454
7-21. H0: =14.25 H1: 14.25
n = 16 = 16.50 s = 5.8 Use = 0.05
t (15) = = 1.55
Right-hand critical point for t (15) and =.05 is 2.131 (and for = .10, t = 1.753).
Do not reject H0 (p-value > .10).
7-22. H0: = 452.8 H1: 452.8
n = 12 = 501.9 s = 65 Use = .05
t (11) = = 2.617
The right-hand critical point for t (11) at = 0.05 is 2.201, and for = 0.02 it is 2.718. Reject H0 at = .05. The p-value is between 0.02 and 0.05.
7-23. H0: = 27 H1: 27 n = 43 s = 4.954 = 24.53 = .05
Critical value for right side: -1.96
z = = -3.269
Reject H0, average staff age is under 27 years old. p-value = .0011
7-24. H0: = 0 H1: 0
n = 13 = 3.1 s =1
t (12) = = 11.177
The p-value is very small. Reject H0. Inclusion in the index likely increases average stock return (at least initially).
EvidenceSample size / 13 / n
Sample Mean / 3.1 / x-bar
Sample Stdev. / 1 / s
s Unknown; Population Normal
Test Statistic / 11.1772 / t
At an a of
Null Hypothesis / p-value / 5%
H0: m = / 0 / 0.0000 / Reject
7-25. H0: = 102.5 H1: 102.5
n = 25 = 107 s = 10 Try = .05 and .01
t (24) = = 2.25
The critical point at = .05 for t (24) is 2.064; the critical point at = .01 = 2.797. Therefore, the results are significant at = .05, and not significant at = .01. Reject H0 at = .05, but not at = .01.
7-26. H0: = 2.5 H1: 2.5
n = 20 = 2.3 = 0.8
Use z because the population is normal and is known.
z = = -1.118
Do not reject H0 (p-value = 0.2636).
7-27. H0: p = .16 H1: p .16
n = 300 x = 51 = 51/300 = .17 use = .05
z = = = 0.472
Do not reject H0 at = .05 (p-value = 0.6370).
z-Test for Population ProportionEvidence / Assumption
Both np and n(1-p) >= 5
Sample size / 300 / n
#Successes / 51 / x
Sample Proportion / 0.1700 / p-hat
Test statistic / 0.4725 / z
At an a of
Hull Hypothesis / p-value / 5%
H0: p = / 0.16 / 0.6366
7-28. H0: p = 0.42 H1: p 0.42
n = 550 x = 219 = 219/550 = 0.39818 use = .01
z = = -1.037
Do not reject H0 (p-value = 0.2998).
7-29. H0: p = 0.12 H1: p 0.12
n = 100 x = 17 use = .05
z = = 1.539
Do not reject H0 (p-value = 0.1238).
7-30. H0: p = 0.35 H1: p 0.35
n = 150 x = 68 = 0.4533 use = .05 and .01
z = = 2.653
Reject H0 at both = .05 and = .01 (p-value = 0.008).
z-Test for Population ProportionEvidence / Assumption
Both np and n(1-p) >= 5
Sample size / 150 / n
#Successes / 68 / x
Sample Proportion / 0.4533 / p-hat
Test statistic / 2.6534 / z
At an a of
Hull Hypothesis / p-value / 5%
H0: p = / 0.35 / 0.0080 / Reject
7-31. H0: p = 0.56 H1: p 0.56
n = 500 x = 298 = 0.596 use = .01
z = = 1.622
Do not reject H0 (p-value = 0.1048).
7-32. H0: u 7% H1: u < 7% n = 41 s = 2.9225 = 2.405 = .01
z = = -10.07
Reject H0, Americans do not save enough.
7-33. H0: = 0 H1: 0
n = 24 = 0.12 s = 0.2 use = .05
t (23) = = 2.939 > 2.069 (the critical value of t (23) for = .05, two-tailed test).
Reject H0. This industry’s stocks probably have positive excess returns in the period in question.
7-34. H0: 344 H1: > 344
n = 1,200 = 361 s = 110
z = = 5.354
Reject H0 (p-value is very small).
EvidenceSample size / 1200 / n
Sample Mean / 361 / x-bar
s Known; Normal Population or Sample Size >= 30
Population Stdev. / 110 / s
Test Statistic / 5.3536 / z
At an a of
Null Hypothesis / p-value / 5%
H0: m = / 344 / 0.0000 / Reject
H0: m ³ / 344 / 1.0000
H0: m £ / 344 / 0.0000 / Reject
7-35. H0: 1.5M H1: > 1.5M
n = 100 = 2.3M s = 0.5M
z = = 16.0
Stongly reject H0 (p-value is very small).
7-36. H0: p 0.17 H1: p > 0.17
n = 2,000 x = 381 = 381/2,000 = 0.1905 use = .01
z = = 2.44 > 2.326
Reject H0 (p-value = .0073). The reported figure of 17% is probably not correct.
Evidence / AssumptionBoth np and n(1-p) >= 5
Sample size / 2000 / n
#Successes / 381 / x
Sample Proportion / 0.1905 / p-hat
Test statistic / 2.4406 / z
At an a of
Hull Hypothesis / p-value / 1%
H0: p = / 0.17 / 0.0147
H0: p >= / 0.17 / 0.9927
H0: p <= / 0.17 / 0.0073 / Reject
7-37. H0: 125 H1: < 125
n = 100 = 121 s = 2
z = = -20
Reject H0 (p-value is very small). Replace all tires.
7-38. Problem 7-38:
(Use template: “testing population mean.xls”; sheet:”sample stats”)
H0: μ =7 H1: μ ≠ 7
Hypothesis Testing - Population Mean
EvidenceSample size / 100 / n
Sample Mean / 4 / x-bar
Sample Stdev. / 3 / s
s Unknown; Population Normal
Test Statistic / -10.0000 / t
At an a of
Null Hypothesis / p-value / 5%
H0: m = / 7 / 0.0000 / Reject
H0: m ³ / 7 / 0.0000 / Reject
H0: m £ / 7 / 1.0000
Reject the null hypothesis: meetings are less than 7 minutes late in starting.
7-39. H0: p .45 H1: p < .45
n = 125 = 49/125 = .392
z = = -1.303
Do not reject H0 (p-value = 0.0962). There is no strong evidence here that the program works in reducing the chance of a midlife crisis.
7-40. H0: p 0.11 H1: p < 0.11
n = 3,500 x = 421 = 421/3,500 = 0.1203
z = = +1.945
Since = .1203 > .11, we need not consider the test statistic value above. Do not reject H0
(p-value = 0.9741 > .05). There is no evidence that the unemployment rate has been reduced.
Evidence / AssumptionBoth np and n(1-p) >= 5
Sample size / 3500 / n
#Successes / 421 / x
Sample Proportion / 0.1203 / p-hat
Test statistic / 1.9448 / z
At an a of
Hull Hypothesis / p-value / 5%
H0: p = / 0.11 / 0.0518
H0: p >= / 0.11 / 0.9741
7-41. H0: 2.5 H1: > 2.5
n = 100 = 5.2 s = 2.8
z = = 9.643
Reject H0 (p-value is very small). There is probable cause for action.
7-42. (Use template: “testing population mean.xls”; sheet:”sample stats”)
H0: μ ≥ 61.4 H1: μ < 61.4
Hypothesis Testing - Population Mean
Sample size / 50 / nSample Mean / 56.8 / x-bar
Sample Stdev. / 10 / s
s Unknown; Population Normal
Test Statistic / -3.2527 / t
At an a of
Null Hypothesis / p-value / 1%
H0: m = / 61.4 / 0.0021 / Reject
H0: m ³ / 61.4 / 0.0010 / Reject
Reject the null hypothesis: the average total private investment per firm is less than $61.4M.
7-43. (Use template: “testing population proportion.xls”; sheet:”normal”)
H0: p ≥ 0.99 H1: p < 0.99
z-Test for Population Proportion
Both np and n(1-p) >= 5
Sample size / 1000 / n
#Successes / 912 / x
Sample Proportion / 0.9120 / p-hat
Test statistic / -24.7900 / z
At an a of
Hull Hypothesis / p-value / 5%
H0: p = / 0.99 / 0.0000 / Reject
H0: p >= / 0.99 / 0.0000 / Reject
H0: p <= / 0.99 / 1.0000
Reject the null hypothesis: the percentage of Swedish customers is significantly less than 99%.
7-44. H0: p = .60 H1: p .60
n = 250 x = 130 = 130/250 = .52 Two-tailed test:
z = = = -2.582
p-value = 2(.5 - .4951) = 2(.0049) = 0.0098. Reject H0 at = .01. The proportion of frequent business travelers who believe that daily service is important has probably decreased.
7-45. Standing start: H0: 5.27 H1: > 5.27
n = 100 = 5.8 s = 1.9